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For example, $P_1^1 = sin\theta$ and $P_2^2 = 3 sin^2\theta$ seem to be not orthogonal to each other because the integral $$\int_0^\pi (sin\theta)(3 sin^2\theta) sin\theta d\theta$$ is not zero. Am I missing something? Thank you for your help!

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    $\begingroup$ You are expecting an orthogonality relation that doesn’t hold. See en.wikipedia.org/wiki/… for the two relations that do hold. See how the two functions have to have the same $l$ or the same $m$? $\endgroup$ – G. Smith Nov 20 '19 at 5:45
  • $\begingroup$ normally we treat the associated legendre functions as polynomials that satisfies $\int_{-1}^1P^m_aP^m_b=C(m,a)\delta_{a,b}$ $\endgroup$ – Ariana Nov 20 '19 at 5:46
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    $\begingroup$ There is no reason to expect them to be orthogonal - they're solutions of different Sturm-Liouville problems, and there is no direct relation between them. $\endgroup$ – Emilio Pisanty Nov 20 '19 at 6:56
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Because the orthogonality is supplied by the azimuthal direction when the two $z$-components of the angular momentum differ. This is why we usually bundle them together into the spherical harmonics instead of handling them separately.

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